Analytic functions of infinite order in half-plane

J. B. Meles (1979) considered entire functions with zeros restricted to a finite number of rays. In particular, it was proved that if f is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity. In this paper, we prove a similar result for a class of functions analytic in the upper half-plane. The analytic function f in C+ = {z : Im z > 0} is called proper analytic if lim sup(z→t)⁡( ln⁡|f(z)|)≤0 for all real numbers t ∈ R. The class of the proper analytic functions is denoted by JA. The full measure of a function f ∈ JA is a positive measure, which justifies the term "proper analytic function". In this paper, we prove that if a function f is the proper analytic function in the half-plane C+ of infinite order with zeros restricted to a finite number of rays Lk through the origin, then its lower order equals infinity.